Harmonic Division

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Harmonic Division∗ Russelle Guadalupe May 10, 2011

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Definition

Let A, B, C, D be four collinear points in this order. The four-point (ACBD) is harmonic division or harmonic if and only if they satisfy AD AC = . CB DB If P is a point not collinear with A, B, C, D, we define a pencil P (ABCD) to be the four lines P A, P B, P C, P D. P (ACBD) is harmonic when (ACBD) is harmonic.

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Theorems 1. If the division (ACBD) is harmonic and O is the midpoint of AB, then OB 2 = OC · OD. 2. If AX, BY, CZ are three concurrent cevians of triangle ABC, with X, Y, Z lying on sides BC, CA, AB respectively, and if ZY ∩ BC = T , then (BXCT ) is harmonic. 3. If four concurrent lines are such that one transversal cuts them harmonically, then every transversal intersects these four lines to form a harmonic division. 4. Let A, B, C, D be four collinear points in this order. If P is a point not collinear with A, B, C, D, then any of the two statements imply the third: a) The division P (ACBD) is harmonic. b) P B bisects ∠CP D internally. c) AP ⊥ P B. 5. If (ACBD) and (A0 C 0 B 0 D0 ) are harmonic, and AA0 , BB 0 , CC 0 concur at P , then D, D0 , P are collinear. 6. Let ABCD be a convex quadrilateral. If K = AD ∩ BC, M = AC ∩ BD, P = AB ∩ KM and Q = DC ∩ KM , then (KM P Q) harmonic. ∗ This

is an unedited version; some theorems are stated without proofs.

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